The Origami Forum

Hello,


I am a paper-folding fan, as well as a Mathematics fan.

I've come up with a way to prove mathematically,
that the square-trisecting fold, as described at http://www.fishgoth.com/origami/diagrams/division.pdf, does in fact deliver the advertised property. My proof is available as an MS-Word document.

I'd appreciate any comments or suggestions.


Thank you,

Itay

P.S.
The proof is available also in Hebrew, if anyone is interested.

you are a pure nerd!!!

can you make a similar method to divide into 5, 7, 11, 13

i am not good at math at all

i working on the crab of lang so 13 is the number i really need

Hello again,


I have been informed, that my proof is not new. As a matter of fact, it appears to be quite famous. Oh, well.


Itay

I'm not very much into math, but I admire you for the effort that you've putted into this


Christian

bacaysao wrote:can you make a similar method to divide into 5, 7, 11, 13

You can make all divisions you like with this theorem:
http://www.origami.gr.jp/People/CAGE_/divide/05-e.html

You can easily generalize Haga's method. If, rather than using 1/2 we use any 0<a<1 as the point to bring the corner to, (a is the distance closer to x on Itay's drawing), then the distance y is easily given by You can easily prove this using basic trigonometry.

I'd like to refer you to Robert Lang's excellent "Origami Constructions" document, which discusses this and other geometric constructions in origami. It can be found on Robert's webpage (http://www.langorigami.com) under science --> Huzita-Hatori axioms.

Thank you, TheRealChris, for your kind words, and thanks everyone else, who responded to my topic.

In addition to http://www.origami.gr.jp/People/CAGE_/d ... dex-e.html , which was mentioned by origami_8, and http://www.langorigami.com, which was mentioned by bshuval, the following internet addresses, related to mathematical paperfolding, have been brought to my attention by Miri Golan and Paul Jackson of the Israeli Origami Center.
1) http://www.math.lsu.edu/~verrill/origami/
2) http://www.msri.org/publications/ln/msr ... index.html
3) http://newmedia.purchase.edu/~Jeanine/origami/

I'd be interested to learn of any other related sites. If more people are interested in this area, it might be worthwhile to compile a comprehensive list of such sites.

Independent discoveries, be it in mathematics or origami design, happen all the time. But it's the journey that's fun, no?

Anyway, I think you know this already, but I'll add Tom Hull's site to this list:

http://www.merrimack.edu/~thull/OrigamiMath.html

He's already compiled a list of a lot of origami mathematics websites out there, and it's kept up to date.

Itay,

If you are willing to also look at books (and articles), not only websites, you will find a wealth of information regarding mathematical paperfolding.

First there is T. Sundara Row's classic "Geometric Exercises in Paperfolding." It can be found in many libraries, and is also rather easy to find to purchase second hand (as I did).

Tom Hull's book "Project Origami" is due to be published by A.K. Peters by the end of the year. I hope it makes it to that date, but I'm doubtful. It is sure to be an indispensable book for any origami-math person.

Speaking of A.K. Peters, they now have a range of origami math books: origami^3 edited by Hull and Origami Design Secrets by Lang. Origami^3 contains the proceedings of the third OSME. It is worth obtaining. Try also to find the proceedings of the first and second OSMEs, but that is going to be more difficult. ODS is Lang's book on origami design.

There are some books in Japanese, too. Haga has his Origamics. Robert Geretschlager (sp?) wrote a manuscript on mathematical paperfolding, which was published in Japanese. Fujimoto's books also look like they contain some origami math, but these are very difficult to obtain and decipher.

This is not an exhaustive list at all. There were many articles published in various levels of accessibility, and one needs only to look hard enough for them. Tom Hull has an excellent bibliography on his website.

Note that not all of these books are accessible for anyone. Some of these get in pretty deep and may bore the average origami enthusiast.